My apologies for delayed response VanHelmontian (By the way are you a fan of his Willow Experiment, the earliest plant physiology test I am aware of?), however I am halfway up a mist shrouded mountain in Catalonia, with intermittent internet access and a less than athletic net book as my only access device.
Nonetheless, I was lucky enough to catch a brief glimpse (45 minutes) of a generously large golden and virtually full Winter Solstice moon, which is a gloriously omnipotent primordial image and relates to both your and other posts with exemplary timing.
The diagram below I've found to be the answer to a moon watchers book of common prayer, despite a convoluted supporting explanation, which I have tried to dissemble into a simpler form that I too can understand as well as everybody else; I class myself as a mere common or back garden scientist.
An important point to note is that exactly one half of the moon is always illuminated by the sun.
Simplified moon phases are as follows: New moon and full moon, first quarter and third quarter, and the phases in between.
Looking at the diagram, we find that the new moon occurs when the moon is positioned between the earth and sun. The three objects are in approximate alignment. The entire illuminated portion of the moon resides on the back side of the moon, the half that is invisible to us.
At a full moon, the earth, moon, and sun are once again in approximate alignment, just as the new moon, but in this case the moon is on the opposite side of the earth, so the entire sunlit part of the moon is facing us and the shadowed portion is entirely hidden from our view.
The first quarter and third quarter moons (both often referred to as a "half moon"), occur when the moon is at a 90 degree angle to both the earth and sun respectively. So we see exactly half of the moon illuminated and half in shadow.
"Between" lunar phase names are identified and more importantly remembered, if it is knowledge you seek, by the use and definition of 4 describing words: crescent, gibbous, waxing, and waning.
The word crescent refers to the phases where the moon is less than half illuminated. The word gibbous refers to phases where the moon is more than half illuminated. Waxing effectively means "growing" or expanding illumination, and waning means "shrinking" or contracting illumination.
Thus you can simply combine the two words to create the phase name, as follows:
After the new moon, the sunlit portion is increasing, but less than half, therefore it is waxing crescent. After the first quarter, the sunlit portion is still increasing, but is now more than half, therefore it is waxing gibbous. After the zenith of a full moon (maximum illumination), the light continually decreases. Thus the waning gibbous phase occurs next. Following the third quarter is the waning crescent, which wanes until the light is completely extinguished and then we arrive at a new moon.